P Versus NP Problem - Formal Definition For NP-completeness

Formal Definition For NP-completeness

There are many equivalent ways of describing NP-completeness.

Let be a language over a finite alphabet .

is NP-complete if, and only if, the following two conditions are satisfied:

  1. ; and
  2. any is polynomial-time-reducible to (written as ), where if, and only if, the following two conditions are satisfied:
    1. There exists such that \forall w\in\Sigma^{*}(w\in L^{'}\Leftrightarrow f(w)\in L); and
    2. there exists a polynomial-time Turing machine that halts with on its tape on any input .

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